Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}-6x+6y &= -8 \\ x+y &= 2\end{align*}$
Begin by moving the $y$ -term in the second equation to the right side of the equation. $x = {-y+2}$ Substitute this expression for $x$ in the first equation. $-6({-y + 2}) + 6y = -8$ $6y - 12 + 6y = -8$ Simplify by combining terms, then solve for $y$ $12y - 12 = -8$ $12y = 4$ $y = \dfrac{1}{3}$ Substitute $\dfrac{1}{3}$ for $y$ in the top equation. $-6x+6( \dfrac{1}{3}) = -8$ $-6x+2 = -8$ $-6x = -10$ $x = \dfrac{5}{3}$ The solution is $\enspace x = \dfrac{5}{3}, \enspace y = \dfrac{1}{3}$.